What function/ functions express radial motion of planet by means of non-linear ODE
$$ \ddot r - \frac{A}{r^3} +\frac{B}{r ^2}=0 $$
(The Kepler/Newton constants are: $\,B= a^3 \omega^2\, ; A=B p \,; $
There is no need to remind.. this differential equation is the, at any rate among the.. oldest of differential equations.
If solutions are elliptic first or second type or both, is the form simplest with Jacobi inverse functions?
EDIT 1:
What is time period in terms of $A,B ?$
EDIT 2:
Also is the Kepler's Equation useful in this closed form derivation?